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Lorenz System

A simplified system of equations describing the 2-D flow of fluid of uniform depth $H$, with an imposed temperature difference $\Delta T$, under gravity $g$, with buoyancy $\alpha$, thermal diffusivity $\kappa$, and kinematic viscosity $\nu$. The full equations are

{\partial\over\partial t}(\nabla^2\phi)={\partial\psi\over\p...
...(\nabla^2\psi)+\nu\nabla^2(\nabla^2\psi) + g\alpha{dT\over dx}
\end{displaymath} (1)

{\partial T\over\partial t}={\partial T\over\partial z}{\par...\nabla^2 T+
{\Delta T\over H}{\partial\psi\over\partial x}.
\end{displaymath} (2)

Here, $\psi$ is the ``stream function,'' as usual defined such that
u={\partial\psi\over\partial x}, \qquad v={\partial\psi\over\partial x}.
\end{displaymath} (3)

In the early 1960s, Lorenz accidentally discovered the chaotic behavior of this system when he found that, for a simplified system, periodic solutions of the form

\psi=\psi_0\sin\left({\pi a x\over H}\right)\sin\left({\pi z\over H}\right)
\end{displaymath} (4)

\theta=\theta_0\cos\left({\pi a x\over H}\right)\sin\left({\pi z\over H}\right)
\end{displaymath} (5)

grew for Rayleigh numbers larger than the critical value, $Ra>Ra_c$. Furthermore, vastly different results were obtained for very small changes in the initial values, representing one of the earliest discoveries of the so-called Butterfly Effect.

Lorenz included the following terms in his system of equations,

$\displaystyle X$ $\textstyle \equiv$ $\displaystyle \psi_{11} \propto \hbox{ convective intensity}$ (6)
$\displaystyle Y$ $\textstyle \equiv$ $\displaystyle T_{11} \propto \Delta T\hbox{ between descending and ascending currents}$ (7)
$\displaystyle Z$ $\textstyle \equiv$ $\displaystyle T_{02} \propto\Delta\hbox{ vertical temperature profile from linearity},$ (8)

and obtained the simplified equations
$\displaystyle \dot X$ $\textstyle =$ $\displaystyle \sigma(Y-X)$ (9)
$\displaystyle \dot Y$ $\textstyle =$ $\displaystyle -XZ+rX-Y$ (10)
$\displaystyle \dot Z$ $\textstyle =$ $\displaystyle XY-bZ,$ (11)

$\displaystyle \sigma$ $\textstyle \equiv$ $\displaystyle {\nu\over\kappa} = \hbox{ Prandtl number}$ (12)
$\displaystyle r$ $\textstyle \equiv$ $\displaystyle {Ra\over Ra_c} = \hbox{ normalized Rayleigh number}$ (13)
$\displaystyle b$ $\textstyle \equiv$ $\displaystyle {4\over 1+a^2} = \hbox{ geometric factor}.$ (14)

Lorenz took $b\equiv{8/3}$ and $\sigma\equiv 10$.

The Critical Points at (0, 0, 0) correspond to no convection, and the Critical Points at

({\sqrt{b(r-1)}, \sqrt{b(r-1)}, r-1})
\end{displaymath} (15)

(-\sqrt{b(r-1)}, -\sqrt{b(r-1)}, r-1)
\end{displaymath} (16)

correspond to steady convection. This pair is stable only if
r = {\sigma(\sigma+b+3)\over \sigma-b-1},
\end{displaymath} (17)

which can hold only for Positive $r$ if $\sigma>b+1$. The Lorenz attractor has a Correlation Exponent of $2.05\pm 0.01$ and Capacity Dimension $2.06\pm 0.01$ (Grassberger and Procaccia 1983). For more details, see Lichtenberg and Lieberman (1983, p. 65) and Tabor (1989, p. 204).

See also Butterfly Effect, Rössler Model


Gleick, J. Chaos: Making a New Science. New York: Penguin Books, pp. 27-31, 1988.

Grassberger, P. and Procaccia, I. ``Measuring the Strangeness of Strange Attractors.'' Physica D 9, 189-208, 1983.

Lichtenberg, A. and Lieberman, M. Regular and Stochastic Motion. New York: Springer-Verlag, 1983.

Lorenz, E. N. ``Deterministic Nonperiodic Flow.'' J. Atmos. Sci. 20, 130-141, 1963.

Peitgen, H.-O.; Jürgens, H.; and Saupe, D. Chaos and Fractals: New Frontiers of Science. New York: Springer-Verlag, pp. 697-708, 1992.

Tabor, M. Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, 1989.

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© 1996-9 Eric W. Weisstein